Stage 3

In many fields, the only way to measure a construct of interest is to have someone produce ratings:

• radiologists’ ratings of disease presence or absence on an X-ray
• researchers rate the amount of bullying occurring in an observed classroom
• coders sort qualitative responses into different response categories

It’s well established in research that multiple raters need to rate the same stimuli to ensure ratings are accurate.  There are a number of ways to measure the agreement among raters using measures of reliability. These differ depending on a host of details, including: the number of raters; whether ratings are nominal, ordinal, or numerical; and whether one rating can be considered a “Gold Standard.”

In this webinar, we will discuss these and other issues in measures of inter and intra rater reliability, the many variations of the Kappa statistic, and Intraclass correlations.

Note: This training is an exclusive benefit to members of the Statistically Speaking Membership Program and part of the Stat’s Amore Trainings Series. Each Stat’s Amore Training is approximately 90 minutes long.

About the Instructor

Audrey Schnell is a statistical consultant and trainer at The Analysis Factor.

Audrey first realized her love for research and, in particular, data analysis in a career move from clinical psychology to research in dementia. As the field of genetic epidemiology and statistical genetics blossomed, Audrey moved into this emerging field and analyzed data on a wide variety of common diseases believed to have a strong genetic component including hypertension, diabetes and psychiatric disorders. She helped develop software to analyze genetic data and taught classes in the US and Europe.

Audrey has worked for Case Western Reserve University, Cedars-Sinai, University of California at San Francisco and Johns Hopkins. Audrey has a Master’s Degree in Clinical Psychology and a Ph.D. in Epidemiology and Biostatistics.

You'll get access to this training webinar, 130+ other stats trainings, a pathway to work through the trainings that you need — plus the expert guidance you need to build statistical skill with live Q&A sessions and an ask-a-mentor forum.

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There are many types and examples of ordinal variables: percentiles, ranks, likert scale items, to name a few.

These are especially hard to know how to analyze–some people treat them as numerical, others emphatically say not to.  Everyone agrees nonparametric tests work, but these are limited to testing only simple hypotheses and designs.  So what do you do if you want to test something more elaborate?

In this webinar we’re going to lay out all the options and when each is (more…)

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Correspondence analysis is a powerful exploratory multivariate technique for categorical variables with many levels. It is a data analysis tool that characterizes associations between levels of two or more categorical variables using graphical representations of the information in a contingency table. It is particularly useful when categorical variables have many levels.

This presentation will give a brief introduction and overview of the use of correspondence analysis, including a review of chi square analysis, and examples interpreting both simple and multiple correspondence plots.

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Smoothing can assist data analysis by highlighting important trends and revealing long term movements in time series that otherwise can be hard to see.

Many data smoothing techniques have been developed, each of which may be useful for particular kinds of data and in specific applications. David will give an introductory overview of the most common smoothing methods, and will show examples of their use. He will cover moving averages, exponential smoothing, the Kalman Filter, low-pass filters, high pass filters, LOWESS and smoothing splines.

This presentation is pitched towards those who may use smoothing techniques during the course of their analytic work, but who have little familiarity with the techniques themselves. David will avoid the underpinning mathematical and statistical methods, but instead will focus on providing a clear understanding of what each technique is about.

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Last year I wrote several articles (GLM in R 1, GLM in R 2, GLM in R 3) that provided an introduction to Generalized Linear Models (GLMs) in R.

As a reminder, Generalized Linear Models are an extension of linear regression models that allow the dependent variable to be non-normal.

In our example for this week we fit a GLM to a set of education-related data.

Let’s read in a data set from an experiment consisting of numeracy test scores (numeracy), scores on an anxiety test (anxiety), and a binary outcome variable (success) that records whether or not the students eventually succeeded in gaining admission to a prestigious university through an admissions test.

We will use the glm() command to run a logistic regression, regressing success on the numeracy and anxiety scores.

A <- structure(list(numeracy = c(6.6, 7.1, 7.3, 7.5, 7.9, 7.9, 8, 
8.2, 8.3, 8.3, 8.4, 8.4, 8.6, 8.7, 8.8, 8.8, 9.1, 9.1, 9.1, 9.3, 
9.5, 9.8, 10.1, 10.5, 10.6, 10.6, 10.6, 10.7, 10.8, 11, 11.1, 
11.2, 11.3, 12, 12.3, 12.4, 12.8, 12.8, 12.9, 13.4, 13.5, 13.6, 
13.8, 14.2, 14.3, 14.5, 14.6, 15, 15.1, 15.7), anxiety = c(13.8, 
14.6, 17.4, 14.9, 13.4, 13.5, 13.8, 16.6, 13.5, 15.7, 13.6, 14, 
16.1, 10.5, 16.9, 17.4, 13.9, 15.8, 16.4, 14.7, 15, 13.3, 10.9, 
12.4, 12.9, 16.6, 16.9, 15.4, 13.1, 17.3, 13.1, 14, 17.7, 10.6, 
14.7, 10.1, 11.6, 14.2, 12.1, 13.9, 11.4, 15.1, 13, 11.3, 11.4, 
10.4, 14.4, 11, 14, 13.4), success = c(0L, 0L, 0L, 1L, 0L, 1L, 
0L, 0L, 1L, 0L, 1L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 
1L, 1L, 1L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L)), .Names = c("numeracy", 
"anxiety", "success"), row.names = c(NA, -50L), class = "data.frame")

attach(A)

names(A)
[1] "numeracy" "anxiety"  "success"

head(A)
    numeracy anxiety  success
1      6.6    13.8       0
2      7.1    14.6       0
3      7.3    17.4       0
4      7.5    14.9       1
5      7.9    13.4       0
6      7.9    13.5       1

The variable ‘success’ is a binary variable that takes the value 1 for individuals who succeeded in gaining admission, and the value 0 for those who did not. Let’s look at the mean values of numeracy and anxiety.

mean(numeracy)
[1] 10.722

mean(anxiety)
[1] 13.954

We begin by fitting a model that includes interactions through the asterisk formula operator. The most commonly used link for binary outcome variables is the logit link, though other links can be used.

model1 <- glm(success ~ numeracy * anxiety, binomial)

glm() is the function that tells R to run a generalized linear model.

Inside the parentheses we give R important information about the model. To the left of the ~ is the dependent variable: success. It must be coded 0 & 1 for glm to read it as binary.

After the ~, we list the two predictor variables. The * indicates that not only do we want each main effect, but we also want an interaction term between numeracy and anxiety.

And finally, after the comma, we specify that the distribution is binomial. The default link function in glm for a binomial outcome variable is the logit. More on that below.

We can access the model output using summary().

summary(model1)
Call:
glm(formula = success ~ numeracy * anxiety, family = binomial)
Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.85712  -0.33055   0.02531   0.34931   2.01048  
Coefficients:
                 Estimate Std. Error z value Pr(>|z|)
(Intercept)       0.87883   46.45256   0.019    0.985
numeracy          1.94556    4.78250   0.407    0.684
anxiety          -0.44580    3.25151  -0.137    0.891
numeracy:anxiety -0.09581    0.33322  -0.288    0.774
(Dispersion parameter for binomial family taken to be 1)
    Null deviance: 68.029  on 49  degrees of freedom
Residual deviance: 28.201  on 46  degrees of freedom
AIC: 36.201

Number of Fisher Scoring iterations: 7

The estimates (coefficients of the predictors – numeracy and anxiety) are now in logits. The coefficient of numeracy is: 1.94556, so that a one unit change in numeracy produces approximately a 1.95 unit change in the log odds (i.e. a 1.95 unit change in the logit).

From the signs of the two predictors, we see that numeracy influences admission positively, but anxiety influences admission negatively.

We can’t tell much more than that as most of us can’t think in terms of logits. Instead we can convert these logits to odds ratios.

We do this by exponentiating each coefficient. (This means raise the value e –approximately 2.72–to the power of the coefficient. e^b).

So, the odds ratio for numeracy is:

OR = exp(1.94556) = 6.997549

However, in this version of the model the estimates are non-significant, and we have a non-significant interaction. Model1 produces the following relationship between the logit (log odds) and the two predictors:

logit(p) = 0.88 + 1.95* numeracy - 0.45 * anxiety - .10* interaction term

The output produced by glm() includes several additional quantities that require discussion.

We see a z value for each estimate. The z value is the Wald statistic that tests the hypothesis that the estimate is zero. The null hypothesis is that the estimate has a normal distribution with mean zero and standard deviation of 1. The quoted p-value, P(>|z|), gives the tail area in a two-tailed test.

For our example, we have a Null Deviance of about 68.03 on 49 degrees of freedom. This value indicates poor fit (a significant difference between fitted values and observed values). Including the independent variables (numeracy and anxiety) decreased the deviance by nearly 40 points on 3 degrees of freedom. The Residual Deviance is 28.2 on 46 degrees of freedom (i.e. a loss of three degrees of freedom).

About the Author: David Lillis has taught R to many researchers and statisticians. His company, Sigma Statistics and Research Limited, provides both on-line instruction and face-to-face workshops on R, and coding services in R. David holds a doctorate in applied statistics.

See our full R Tutorial Series and other blog posts regarding R programming.

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Latent Class Analysis is a method for finding and measuring unobserved latent subgroups in a population based on responses to a set of observed categorical variables.

This webinar will present an overview and an example of how latent class analysis works to find subgroups, how to interpret the output, the steps involved in running it.  We will discuss extensions and uses of the latent classes in other analyses and similarities and differences with related techniques.

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